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28
Unification: A multidisciplinary survey
 ACM Computing Surveys
, 1989
"... The unification problem and several variants are presented. Various algorithms and data structures are discussed. Research on unification arising in several areas of computer science is surveyed, these areas include theorem proving, logic programming, and natural language processing. Sections of the ..."
Abstract

Cited by 103 (0 self)
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The unification problem and several variants are presented. Various algorithms and data structures are discussed. Research on unification arising in several areas of computer science is surveyed, these areas include theorem proving, logic programming, and natural language processing. Sections of the paper include examples that highlight particular uses
Theorem Proving with Ordering and Equality Constrained Clauses
 Journal of Symbolic Computation
, 1995
"... constraint strategies and saturation Given a signature F , below we denote by S the set of all clauses built over F , and similarly by C the set of all constraints, and by EC the set of all equality constraints (which is a subset of C). Definition 3.1. An inference rule IR is a mapping of ntuples ..."
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Cited by 74 (19 self)
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constraint strategies and saturation Given a signature F , below we denote by S the set of all clauses built over F , and similarly by C the set of all constraints, and by EC the set of all equality constraints (which is a subset of C). Definition 3.1. An inference rule IR is a mapping of ntuples of clauses to sets of triples containing a clause, a constraint and an equality constraint: IR : S n \Gamma! P(hS; C; ECi) An inference system is a set of inference rules. Definition 3.2. A constraint inheritance strategy is a function mapping a clause, two constraints and an equality constraint to a clause and a constraint: H : S \Theta C \Theta C \Theta EC \Gamma! S \Theta C Inference systems and constraint inheritance strategies are combined to produce inferences in the usual sense: given constrained clauses C 1 [[T 1 ]]; : : : ; Cn [[T n ]], we obtain a conclusion C [[T ]] as follows. Applying an inference rule to C 1 ; : : : ; Cn we obtain a triple hD; OT;ET i. Then the constraint...
HigherOrder Tableaux
, 1995
"... Even though higherorder calculi for automated theorem proving are rather old, tableau calculi have not been investigated yet. This paper presents two free variable tableau calculi for higherorder logic that use higherorder unification as the key inference procedure. These calculi differ in the ..."
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Cited by 16 (6 self)
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Even though higherorder calculi for automated theorem proving are rather old, tableau calculi have not been investigated yet. This paper presents two free variable tableau calculi for higherorder logic that use higherorder unification as the key inference procedure. These calculi differ in the treatment of the substitutional properties of equivalences. The first calculus is equivalent in deductive power to the machineoriented higherorder refutation calculi known from the literature, whereas the second is complete with respect to Henkin's general models.
Extensional higherorder paramodulation and RUEresolution
 AUTOMATED DEDUCTION — CADE16 INTERNATIONAL CONFERENCE, LNAI 1632
, 1999
"... This paper presents two approaches to primitive equality treatment in higherorder (HO) automated theorem proving: a calculus EP adapting traditional firstorder (FO) paramodulation [RW69] , and acalculusERUE adapting FO RUEResolution [Dig79] to classical type theory, i.e., HO logic based on Church ..."
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Cited by 14 (8 self)
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This paper presents two approaches to primitive equality treatment in higherorder (HO) automated theorem proving: a calculus EP adapting traditional firstorder (FO) paramodulation [RW69] , and acalculusERUE adapting FO RUEResolution [Dig79] to classical type theory, i.e., HO logic based on Church’s simply typed λcalculus. EP and ERUE extend the extensional HO resolution approach ER [BK98a]. In order to reach Henkin completeness without the need for additional extensionality axioms both calculi employ new, positive extensionality rules analogously to the respective negative ones provided by ER that operate on unification constraints. As the extensionality rules have an intrinsic and unavoidable differencereducing character the HO paramodulation approach loses its pure termrewriting character. On the other hand examples demonstrate that the extensionality rules harmonise quite well with the differencereducing HO RUEresolution idea.
ProofTerm Synthesis on Dependenttype Systems via Explicit Substitutions
, 1999
"... Typed #terms are used as a compact and linear representation of proofs in intuitionistic logic. This is possible since the CurryHoward isomorphism relates proof trees with typed #terms. The proofsasterms principle can be used to check a proof by type checking the #term extracted from the compl ..."
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Cited by 8 (1 self)
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Typed #terms are used as a compact and linear representation of proofs in intuitionistic logic. This is possible since the CurryHoward isomorphism relates proof trees with typed #terms. The proofsasterms principle can be used to check a proof by type checking the #term extracted from the complete proof tree. However, proof trees and typed #terms are built differently. Usually, an auxiliary representation of unfinished proofs is needed, where type checking is possible only on complete proofs. In this paper we present a proof synthesis method for dependenttype systems where typed open terms are built incrementally at the same time as proofs are done. This way, every construction step, not just the last one, may be type checked. The method is based on a suitable calculus where substitutions as well as metavariables are firstclass objects.
A Calculus and a System Architecture for Extensional HigherOrder Resolution
, 1997
"... The first part of this paper introduces an extension for a variant of Huet's higherorder resolution calculus [Hue72, Hue73] based upon classical type theory (Church's typed calculus [Chu40]) in order to obtain a calculus which is complete with respect to Henkin models [Hen50]. The new rules connec ..."
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Cited by 8 (5 self)
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The first part of this paper introduces an extension for a variant of Huet's higherorder resolution calculus [Hue72, Hue73] based upon classical type theory (Church's typed calculus [Chu40]) in order to obtain a calculus which is complete with respect to Henkin models [Hen50]. The new rules connect higherorder preunification with the general refutation process in an appropriate way to establish full extensionality for the whole system. The general idea of the calculus is discussed on different examples. The second part introduces the Leo system which implements the discussed extensional higherorder resolution calculus. This part mainly focus on the embedding of the new extensionality rules into the refutation process and the treatment of higherorder unification. 1 Introduction Many mathematical problems can be expressed shortly and elegantly in higher order logic whereas they often lead to unnatural and inflated formulations in firstorder logic, e.g., when coding them into axio...
On The Use Of Constraints In Automated Deduction
, 1995
"... . This paper presents three approaches dealing with constraints in automated deduction. Each of them illustrates a different point. The expression of strategies using constraints is shown through the example of a completion process using ordered and basic strategies. The schematization of complex un ..."
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Cited by 7 (1 self)
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. This paper presents three approaches dealing with constraints in automated deduction. Each of them illustrates a different point. The expression of strategies using constraints is shown through the example of a completion process using ordered and basic strategies. The schematization of complex unification problems through constraints is illustrated by the example of an equational theorem prover with associativity and commutativity axioms. The incorporation of builtin theories in a deduction process is done for a narrowing process which solves queries in theories defined by rewrite rules with builtin constraints. Advantages of using constraints in automated deduction are emphasized and new challenging problems in this area are pointed out. 1 Motivations Constraints have been introduced in automated deduction since about 1990, although one could find similar ideas in theory resolution [32] and in higherorder resolution [16]. The idea is to distinguish two levels of deduction and t...
On the Translation of HigherOrder Problems into FirstOrder Logic
 Proceedings of ECAI94
, 1994
"... . In most cases higherorder logic is based on the  calculus in order to avoid the infinite set of socalled comprehension axioms. However, there is a price to be paid, namely an undecidable unification algorithm. If we do not use the calculus, but translate higherorder expressions into firstor ..."
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Cited by 6 (4 self)
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. In most cases higherorder logic is based on the  calculus in order to avoid the infinite set of socalled comprehension axioms. However, there is a price to be paid, namely an undecidable unification algorithm. If we do not use the calculus, but translate higherorder expressions into firstorder expressions by standard translation techniques, we have to translate the infinite set of comprehension axioms, too. Of course, in general this is not practicable. Therefore such an approach requires some restrictions such as the choice of the necessary axioms by a human user or the restriction to certain problem classes. This paper will show how the infinite class of comprehension axioms can be represented by a finite subclass, so that an automatic translation of finite higherorder problems into finite firstorder problems is possible. This translation is sound and complete with respect to a Henkinstyle general model semantics. 1 Introduction Firstorder logic is a powerful tool for ...
Generation as Deduction on Labelled Proof Nets
 Logical Aspects of Computational Linguistics, LACL’96, volume 1328 of Lecture Notes in Artificial Intelligence
"... . In the framework of labelled proof nets the task of parsing in categorial grammar can be reduced to the problem of firstorder matching under theory. Here we shall show how to use the same method of labelled proof nets to reduce the task of generating to the problem of higherorder matching. 1 Int ..."
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Cited by 5 (1 self)
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. In the framework of labelled proof nets the task of parsing in categorial grammar can be reduced to the problem of firstorder matching under theory. Here we shall show how to use the same method of labelled proof nets to reduce the task of generating to the problem of higherorder matching. 1 Introduction Categorial grammar provides a mechanism for the analysis of linguistic expressions on the basis of lexicalism and the parsing as deduction paradigm ([17]). 3 In accordance with lexicalism each lexical entry of the language encapsulates all the information needed to analyse the lexical item, and the grammar itself only needs to know how to manage these resources. In the particular case of categorial grammar, a lexical categorisation is a formula, or type, constructed over some basic types by logical connectives; and the grammar constitutes the connectives' syntactic behaviour (i.e. the laws governing the connectives). Within the parsing as deduction paradigm the problem of analys...